An alternative way to derive Theorem 9.5.1 uses the following division rule: Let n and k be integers so that k divides n. If a set consisting of n elements is divided into subsets that each contain k elements, then the number of such subsets is n/k. Explain how Theorem 9.5.1 can be derived using the division rule.
The theorem using the division rule.
Division Rule: Let be integers so that divides . If a set consisting of elements is divided into subsets that each contains elements, then the number of such subsets is .
Now, consider a set of elements, out of which a subset of size r is required to calculate.
Now, consider mapping any permutation of an -element set into -element set.
Such mapping is done by taking the first elements of the permutation.
Notice that any other permutation with the same first elements in any order and the same remaining elements in any order will also map to this set.
Now there are possible permutations of the first elements.
Also permutations are possible for the remaining elements
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started